Nikica Jelavic opened the scoring after 12 minutes and Leighton Baines doubled Everton's lead seven minutes later from the penalty spot.
Leon Osman made it 3-0 early in the second period and, after Russell Penn scored pulled one back, Seamus Coleman and Marouane Fellaini completed Everton's nap hand.

David Moyes' side will now visit the winners of next week's Sunderland-Bolton replay at the Stadium Of Light in the fourth round.

Moyes paid Cheltenham the compliment of parading eight of the line-up which started the 2-1 midweek win at Newcastle in the Premier League.
Their decisive start to the game, coupled with the powerful nature of the team, ensured there was little chance of a shock result.

Fellaini was a constant threat in a free role while full-backs Coleman and Baines got forward to good effect.
But promotion-chasing Cheltenham showed plenty of fight with Penn and Darren Carter impressing in midfield.

Cheltenham started brightly and former Birmingham player Carter had half a chance but dragged his shot wide after being found in space by Kaid Mohamed.
Jelavic teed up Bryan Oviedo on the edge of the Robins box but his low drive did not extend keeper Scott Brown.

However, Everton broke the deadlock after 12 minutes through Jelavic.
Fellaini's curling shot from the right of the Cheltenham box contained too much power for Brown as it crashed against the far post.
But Jelavic was alert to the rebound and made no mistake for his seventh goal of the campaign.

Jelavic shot wide after more good play by Fellaini and Brown produced a superb save to keep out Baines' curling free-kick.

Carter was looking sharp for Cheltenham and forced a low save out of Tim Howard.
But after 19 minutes Baines doubled the Toffees lead from the penalty spot.
Cheltenham skipper Alan Bennett was adjudged to have fouled Fellaini and the resulting spot-kick from Baines contained too much power for Brown even though he dived the right way.

Fellaini was inches wide with a low shot on the turn as Everton threatened danger every time they attacked.
But Cheltenham were showing plenty of spirit and Jermaine McGlashan volleyed over from inside the box.

Fellaini got into another good position inside the six yard box but glanced his header wide from Coleman's centre.
The impressive Osman appeared to have killed off the tie after 49 minutes when he slotted the ball past Brown after taking a Baines pass in his stride.

Cheltenham pulled one back two minutes later through Penn's cool finish after good play by Mohamed.
But Coleman re-established Everton's three goal lead after 57 minutes.
He burst into the box and dinked the ball over Brown after exchanging passes with Victor Anichebe.

With the tie safely secured, Moyes was able to bring off Baines, Osman and Jelavic and replace them with Magaye Gueye, Steven Naismith and Thomas Hitzlsperger.

Cheltenham continued to battle away and Steve Elliott was not far off target with a powerful header.
But it needed a fine save from Brown to deny Naismith before Fellaini made it 5-1 with a minute left from Anichebe's cross.